Magnetic-field-dependent stimulated emission from nitrogen-vacancy centers in diamond

Negatively charged nitrogen-vacancy (NV) centers in diamond are promising magnetic field quantum sensors. Laser threshold magnetometry theory predicts improved NV center ensemble sensitivity via increased signal strength and magnetic field contrast. Here, we experimentally demonstrate laser threshold magnetometry. We use a macroscopic high-finesse laser cavity containing a highly NV-doped and low absorbing diamond gain medium that is pumped at 532 nm and resonantly seeded at 710 nm. This enables a 64% signal power amplification by stimulated emission. We test the magnetic field dependency of the amplification and thus demonstrate magnetic field–dependent stimulated emission from an NV center ensemble. This emission shows an ultrahigh contrast of 33% and a maximum output power in the milliwatt regime. The coherent readout of NV centers pave the way for novel cavity and laser applications of quantum defects and diamond NV magnetic field sensors with substantially improved sensitivity for the health, research, and mining sectors.


S1. Investigation of the cavity parameters
We investigate the origin of the amplification by the detection of the cavity amplitude, FSR and FWHM during a scan of the pump power in Fig. S1. The measurement proves that the increase of the cavity finesse (orange trace) comes from gain caused by stimulated emission reducing the total loss of the cavity (blue trace). The gain shown in the main article is calculated from the finesse with formula µ g = π/l(F −1 0 − F −1 g ). The value of the finesse at zero pump power is F 0 = 847 and the maximal value when pumping is F max = 1011. The FSR is constant as expected. This shows that no thermal effect, like thermal expansion of the diamond is influencing the FSR of the cavity. Stimulated emission, should also lead to an increasing number of photons n in the cavity, not just a narrowing of the peak width. Thus, an increase in the area S ∝ n is expected, as well. The Lorentzian amplitude A is given by A = 2S/(πµ) with the area S under the curve and the total gain/loss µ = µ 0 L − µ g l. Thus, the area is calculated from the measured amplitude and FWHM and shown in Fig. S1. The increase in the area proves the increase in the number of photons in the cavity as expected from stimulated emission.

S2. Magnetic field of the permanent magnet
We measured the magnetic field with a Hall sensor at varying distance from the magnet. The magnetic field is applied in the (100) crystal direction. The diamond contains NV-centres in all four possible NV-directions. In this configuration the projections of the applied magnetic field onto and perpendicular to the NV-axis is symmetric and of the same magnitude for all NV-orientations. The angle between the NV-axis is α = 109.5 • (21). The effective transverse magnetic field that the electron spin is interacting with is then calculated by B x = B sin(α/2), where B is the magnitude of the measured magnetic field. We assume a constant magnetic field within the mode of the diamond of diameter 2ω 0 ≈ 110 µm. The resulting effective magnetic field is shown in Fig. S2. The excited NV-centres in the diamond mode in the experiment are located at a distance of (2 ± 1) mm from the magnet. The effective transverse magnetic field is S3. NV PL contrast by a constant transverse magnetic field B x Stimulated emission in a cavity can significantly enhance the contrast, theoretically up to 100 % when working above a laser threshold (22). The strong contrast observed in the main article in stimulated emission is higher than the maximal contrast possible in spontaneous emission according to the current model of the NV centre, as we show below. We calculate theoretically the change of spontaneous emission, i.e. the photoluminescence contrast of the NV-centres as a function of a constant, external magnetic field perpendicular to the NV direction. Without loss of generality, the perpendicular magnetic field can be chosen to the match the direction of transverse magnetic field component B x and leads to mixing spin states. We assume the 7-level model of Fig. S3A for the calculation. Following established litera-ture (21), we assume the radiative decay rate Γ = 83.3 MHz, the intersystem crossing rate from the ±1 states to the singlet state L 1 = 1/(2 * 12 ns + 0.9 ns) as half of the radiative decay rate, no transition from the m s = 0 state to the singlet state, the rate L 3 = 2.2 MHz and L 2 = L 3 /2, such that the lifetime of the singlet ground state becomes 1/(L 2 + L 3 ) = 308 ns. The singlet state is just represented by one state. However, it has an additional short-lived excited state with a lifetime ≈ 0.9 ns (21), which we integrated into the rate L 1 to simplify the calculation.
Assuming the weak excitation limit, we consider the effect of magnetic field on the groundstate spin triplet only, and ignore dynamics and coherence in the excited-state triplet. The ground state triplet with spin operators ⃗ S in an external magnetic field ⃗ B is represented by the Hamiltonian where B c = (B x + iB y )/ √ 2 represents the magnetic field perpendicular to the NV direction, We find that the NV emission is reduced by a transverse magnetic field as shown in Fig. S3B.
This relative reduction is true for all excitation rates Λ ≪ Γ in the weak-excitation regime, for which this solution is valid. Since we are pumping a cavity mode with 111 µm diameter even a high pump power of 3 W is still in the weak-excitation regime Λ < 0.1Γ and far from saturation, see Fig. S5. The magnetic field along the NV direction B z = 0 in Fig. S3B. Minimizing the relative emission for arbitrary B z and B c we find that the minimum is 0.78, i.e. the strongest reduction is 22%.
This means that the maximal contrast of spontaneous emission of an NV ensemble, caused by the introduction of an external magnetic field is 22 %. The literature parameters of the model (21) were developed from mostly measurements of ensembles of NV centres, so the result applies to ensembles and single NV centres can potentially achieve better contrast. Specifically, single NV-centres in a confocal setup can be pumped into a high-excitation limit, which can enhance contrast to >40 % when a permanent magnetic field mixes the ground-and excitedstate spin populations (42). However, for pumping larger volumes of NV ensembles even high powers typically stay in the weak-excitation limit, as is the case for all our measurements.
The strong contrast in ensembles measured through stimulated emission in the main article is stronger than a contrast could be with spontaneous emission according to the current model for NV centre ensembles.

S4. Amplification and absorption study of different samples
In Fig. S4A the relative amplification, absorption at the seeding wavelength of 710 nm and NV − concentration are shown for four different samples that were investigated. The data shows that both, low absorption and high NV − concentration, i. e. high gain, is needed for a strong stimulated emission signal. For decreasing absorption the amplification is increasing, as the loss inside the cavity is decreasing. At the same time the amplification is also dependent on the gain which is proportional to the number of pumped NV-centres (37). The highest amplification Magnetic field B x (mT) The energy differences are taken from (43). The population of the third phonon level is below 1 %, consequently the absorption of the red seeding laser at this transition is negligible.

S5. Saturation measurement -PL calibration
We collected the PL with an objective laterally to the pumped NV centres (main article, Fig. 1B) and measured the power with a photo-diode power sensor (det3). The PL was maximised for this measurement. We fitted the function P PL = P inf P pump /(P pump + P sat ) to the data with the fitting parameters P inf , P sat . The fit gave P inf = 587.68 µW which is the PL power at infinite pump power and P sat = 70.61 W which is the pump power at saturation. In the ODMR measurement

S6. ODMR magnetic field sensitivity
In measurements of the magnetic field B in the standard quantum limit the measurement error, i.e. the standard deviation σ B ∝ 1/ √ T , reduces with longer measurement time T. Therefore, the definition of the DC magnetic field sensitivity captures precision independent of the measurement time and is defined as η DC = σ B √ T . Assuming photon shot-noise limited sensitivity, it can be calculated directly from the intensity I in counts per second as In the second step the optically detected magnetic resonance (ODMR) measurement is con- 3.75 W , the seeding power was 1.32 W. This regime was not optimal for the stimulated emission contrast since the strong seeding laser is not filtered from the signal but achieved much higher output powers and thus improved sensitivity due to stronger stimulated emission from the increased seeding. A double Lorentzian is fitted to the data in the main article (Fig. 5). Here ν, P 0 , C, ∆ν, ν c are the microwave frequency, baseline, contrast, HWHM and centre frequency, respectively. The index represents the number of the single Lorentzian. The data is shown in Fig. S6. The plots also show the contributions of the single Lorentzian fits. The related fit parameters are listed in Table S1. The parameters for the calculation of the setup sensitivities were taken from column C, index 2 for the PL measurement and from column D, index 2 for the cavity setup.     stimulated emission is below the temporal resolution < 1 ms, limited due to the unlocked cavity setup and read-out of the cavity amplitude. The amplitude also immediately decays when the pump laser is turned off, as expected for stimulated emission. However, when the pump is turned off the decay of the cavity amplitude goes below the initial value before pumping. This means less red light is transmitted through the cavity after the green laser illumination than before. In other word an additional absorption at the seed wavelength of 710 nm is 'induced' by the green pump laser. After the green laser is turned off and the signal has immediately dropped to a value lower than the initial value, the signal starts to rise again slowly back to the initial value before green pumping. The decay time of the "induced absorption" is in the order of several seconds τ decay = 5.7 s. The temporal behaviour indicates that we are not in the regime of excited state absorption due to the NV − excited state lifetime of 12 ns (21).

S7. Ionisation and induced absorption
However, ionisation occurs with increasing pump power as shown in Fig. S7A. A certain reduction of the NV-population can be seen although it is still above 80 % of the initial value for pump powers below 5W. While this could explain a small reduction in gain from NV-centres, it can not be the reason for the induced absorption of the red laser since the created NV 0 centres do not absorb light at 710 nm. We think the ionisation plays a different role in this novel effect.
We suggest the following explanation: The photoionisation of the NV centres leads the creation of certain charge states of other defects or in other words leads to a population of long lived intra band gap states. These states absorb light at the seeding wavelength of 710 nm. A possible candidate could be the H2/H3 centre in diamond as the H2 centre absorbs at this wavelength (44).
The resulting absorption induced by the pump laser is higher than the initial absorption before the NV centres were pumped. The red seeding is kept constant at P seed = 610 mW. After 10 s the pump laser is applied at a pump power of P 532 = 260 mW and turned off after another 30 s. An exponential function y = a * exp(−t/τ decay ) + b is fitted to the slow decay when the pump laser is turned off.

S8. Analytic solution of an externally seeded NV cavity
For the seeded NV lasing cavity we assume the 7-level model presented in Fig. S8A with a constant excitation rate Λ and coupling constant between the NV centres and the cavity G for both spin states. We set up the equations of motion describing the populations of the 7-level system as a superoperator matrix and assume the transition rates by the standard NV model as in (21,22). describes the seeded cavity with seeding rate α and number of cavity photons per NV centre n.
In order to find the solution of the system we first calculate the steady state solution, i.e. the nullspace of the superoperator matrix, treating n as a steady-state parameter. We solve for two cases: no applied magnetic field B x = 0 and large magnetic field γ e B x ≫ D g , where the spin states are fully mixed by the transverse magnetic field. We normalise the resulting solutions for the density matrix elements ρ ij via the sum of the diagonal elements ρ ii and insert these into the cavity rate equation (S.5) for the steady state to find the non-zero solution for n ̸ = 0.
We then simulate the behaviour of the system with an applied transverse magnetic field B x via the stimulated emission rate of the cavity by Γ stim ∝ ρ e Gn, where the excited state population is ρ e = ρ 22 + ρ 55 − ρ 33 − ρ 66 .
The simulation in the main article in Fig. 3 calculates the contrast by stimulated emission via the solution at zero magnetic field B x = 0 and strong magnetic field, i.e. D g ≪ γ e B x / √ 2. We assume a constant excitation rate Λ = 1 MHz in the low excitation regime leading to a cavity below threshold as in the experiment. The simulation in Fig. S8B shows an increasing contrast by stimulated emission when the seeding is low, i.e. the contribution of the stimulated emission to the overall cavity signal is large. This is what is also measured experimentally in the main article in Fig. 4C and Fig. 5A. In this regime the contrast by stimulated emission exceeds the maximal achievable contrast via PL of 22%. The seven states are numbered by |X⟩ and the transition rates between the states by L XX . Stimulated emission is generated by the coupling between the NV and the cavity G. The ground state is split by the zero field splitting constant D g . The dephasing rate between the ground spin sates is given by Γ 14 . The applied transverse magnetic field B x leads to spin-mixing similar to the PL case in Fig. S3A.
in the rotating wave approximation (RWA) describes the system.
We can now compare the Hamiltonian for a constant transverse magnetic field above (Ch.S3, supplementary material) with the Hamiltonian for a constant microwave (MW) drive H M W .
For both experimental methods in the main article the component B z = 0, as the applied field is transverse (method 1) and we don't apply a bias field (method 2).
We now distinguish two cases: 1) The case without spin mixing, i.e. a zero transverse field B x = 0 (method 1) corresponding to a detuned microwave drive ω ≪ D g or ω ≫ D g (method 2) and 2) the case with spin mixing, i.e. a strong transverse magnetic γ e B x ≫ D g (method 1) corresponding to a resonant drive ω = D g (method 2). For both cases the Hamiltonians H P L , H M W are mathematically equivalent and the physics of the NV centre is the same. Of course the microwave field B 1 mixes the spin states at much lower field values than a transverse magnetic field, since the transverse field B x has to overcome the zero-field splitting D g . However this shows that in the extreme cases the transverse magnetic field component B x of the permanent magnet plays the equivalent role to the constant microwave amplitude B 1 for mixing the spin states.
In the experiment a transverse magnetic field is the easiest way to test maximal spin mixing due to better homogeneity. Once an improved contrast is measured that way (see main article) we succeeded to see this improved contrast in ODMR as well, which is lower to the permanent magnet as it has the additional challenge of creating a good homogeneity but is the necessary way for good sensitivity.